Changeset 11597 for NEMO/trunk/doc/latex/NEMO/subfiles/apdx_invariants.tex
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- 2019-09-25T20:20:19+02:00 (5 years ago)
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NEMO/trunk/doc/latex/NEMO/subfiles/apdx_invariants.tex
r11596 r11597 12 12 %\gmcomment{ 13 13 14 %% ================================================================================================= 14 15 \section{Introduction / Notations} 15 16 \label{sec:INVARIANTS_0} … … 85 86 \end{flalign} 86 87 88 %% ================================================================================================= 87 89 \section{Continuous conservation} 88 90 \label{sec:INVARIANTS_1} … … 311 313 % 312 314 315 %% ================================================================================================= 313 316 \section{Discrete total energy conservation: vector invariant form} 314 317 \label{sec:INVARIANTS_2} 315 318 319 %% ================================================================================================= 316 320 \subsection{Total energy conservation} 317 321 \label{subsec:INVARIANTS_KE+PE_vect} … … 337 341 leads to the discrete equivalent of the four equations \autoref{eq:INVARIANTS_E_tot_flux}. 338 342 343 %% ================================================================================================= 339 344 \subsection{Vorticity term (coriolis + vorticity part of the advection)} 340 345 \label{subsec:INVARIANTS_vor} … … 343 348 or the planetary ($q=f/e_{3f}$), or the total potential vorticity ($q=(\zeta +f) /e_{3f}$). 344 349 Two discretisation of the vorticity term (ENE and EEN) allows the conservation of the kinetic energy. 350 %% ================================================================================================= 345 351 \subsubsection{Vorticity term with ENE scheme (\protect\np[=.true.]{ln_dynvor_ene}{ln\_dynvor\_ene})} 346 352 \label{subsec:INVARIANTS_vorENE} … … 380 386 In other words, the domain averaged kinetic energy does not change due to the vorticity term. 381 387 388 %% ================================================================================================= 382 389 \subsubsection{Vorticity term with EEN scheme (\protect\np[=.true.]{ln_dynvor_een}{ln\_dynvor\_een})} 383 390 \label{subsec:INVARIANTS_vorEEN_vect} … … 449 456 \end{flalign*} 450 457 458 %% ================================================================================================= 451 459 \subsubsection{Gradient of kinetic energy / Vertical advection} 452 460 \label{subsec:INVARIANTS_zad} … … 556 564 Blah blah required on the the step representation of bottom topography..... 557 565 566 %% ================================================================================================= 558 567 \subsection{Pressure gradient term} 559 568 \label{subsec:INVARIANTS_2.6} … … 698 707 Nevertheless, it is almost never satisfied since a linear equation of state is rarely used. 699 708 709 %% ================================================================================================= 700 710 \section{Discrete total energy conservation: flux form} 701 711 \label{sec:INVARIANTS_3} 702 712 713 %% ================================================================================================= 703 714 \subsection{Total energy conservation} 704 715 \label{subsec:INVARIANTS_KE+PE_flux} … … 721 732 vector invariant or in flux form, leads to the discrete equivalent of the ???? 722 733 734 %% ================================================================================================= 723 735 \subsection{Coriolis and advection terms: flux form} 724 736 \label{subsec:INVARIANTS_3.2} 725 737 738 %% ================================================================================================= 726 739 \subsubsection{Coriolis plus ``metric'' term} 727 740 \label{subsec:INVARIANTS_3.3} … … 742 755 The derivation is the same as for the vorticity term in the vector invariant form (\autoref{subsec:INVARIANTS_vor}). 743 756 757 %% ================================================================================================= 744 758 \subsubsection{Flux form advection} 745 759 \label{subsec:INVARIANTS_3.4} … … 820 834 The horizontal kinetic energy is not conserved, but forced to decay (\ie\ the scheme is diffusive). 821 835 836 %% ================================================================================================= 822 837 \section{Discrete enstrophy conservation} 823 838 \label{sec:INVARIANTS_4} 824 839 840 %% ================================================================================================= 825 841 \subsubsection{Vorticity term with ENS scheme (\protect\np[=.true.]{ln_dynvor_ens}{ln\_dynvor\_ens})} 826 842 \label{subsec:INVARIANTS_vorENS} … … 889 905 The later equality is obtain only when the flow is horizontally non-divergent, \ie\ $\chi$=$0$. 890 906 907 %% ================================================================================================= 891 908 \subsubsection{Vorticity Term with EEN scheme (\protect\np[=.true.]{ln_dynvor_een}{ln\_dynvor\_een})} 892 909 \label{subsec:INVARIANTS_vorEEN} … … 959 976 \end{flalign*} 960 977 978 %% ================================================================================================= 961 979 \section{Conservation properties on tracers} 962 980 \label{sec:INVARIANTS_5} … … 972 990 as the equation of state is non linear with respect to $T$ and $S$. 973 991 In practice, the mass is conserved to a very high accuracy. 992 %% ================================================================================================= 974 993 \subsection{Advection term} 975 994 \label{subsec:INVARIANTS_5.1} … … 1035 1054 which is the discrete form of $ \frac{1}{2} \int_D { T^2 \frac{1}{e_3} \frac{\partial e_3 }{\partial t} \;dv }$. 1036 1055 1056 %% ================================================================================================= 1037 1057 \section{Conservation properties on lateral momentum physics} 1038 1058 \label{sec:INVARIANTS_dynldf_properties} … … 1053 1073 the term associated with the horizontal gradient of the divergence is locally zero. 1054 1074 1075 %% ================================================================================================= 1055 1076 \subsection{Conservation of potential vorticity} 1056 1077 \label{subsec:INVARIANTS_6.1} … … 1084 1105 \end{flalign*} 1085 1106 1107 %% ================================================================================================= 1086 1108 \subsection{Dissipation of horizontal kinetic energy} 1087 1109 \label{subsec:INVARIANTS_6.2} … … 1133 1155 \] 1134 1156 1157 %% ================================================================================================= 1135 1158 \subsection{Dissipation of enstrophy} 1136 1159 \label{subsec:INVARIANTS_6.3} … … 1154 1177 \end{flalign*} 1155 1178 1179 %% ================================================================================================= 1156 1180 \subsection{Conservation of horizontal divergence} 1157 1181 \label{subsec:INVARIANTS_6.4} … … 1178 1202 \end{flalign*} 1179 1203 1204 %% ================================================================================================= 1180 1205 \subsection{Dissipation of horizontal divergence variance} 1181 1206 \label{subsec:INVARIANTS_6.5} … … 1201 1226 \end{flalign*} 1202 1227 1228 %% ================================================================================================= 1203 1229 \section{Conservation properties on vertical momentum physics} 1204 1230 \label{sec:INVARIANTS_7} … … 1369 1395 \end{flalign*} 1370 1396 1397 %% ================================================================================================= 1371 1398 \section{Conservation properties on tracer physics} 1372 1399 \label{sec:INVARIANTS_8} … … 1378 1405 As for the advection term, there is conservation of mass only if the Equation Of Seawater is linear. 1379 1406 1407 %% ================================================================================================= 1380 1408 \subsection{Conservation of tracers} 1381 1409 \label{subsec:INVARIANTS_8.1} … … 1408 1436 In fact, this property simply results from the flux form of the operator. 1409 1437 1438 %% ================================================================================================= 1410 1439 \subsection{Dissipation of tracer variance} 1411 1440 \label{subsec:INVARIANTS_8.2}
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